Abstract

We demonstrate in both simulated and real cases the effect that undersampling of a three-dimensional (3D) wrapped phase distribution has on the geometry of phase singularity loops and their branch cut surfaces. The more intuitive two-dimensional (2D) problem of setting branch cuts between dipole pairs is taken as a starting point, and then branch cut surfaces in flat and ambiguous 3D loops are discussed. It is shown that the correct 2D branch cuts and 3D branch cut surfaces should be placed where the gradient of the original phase distribution exceeded π rad voxel−1. This information, however, is lost owing to undersampling and cannot be recovered from the sampled wrapped phase distribution alone. As a consequence, empirical rules such as finding the surface of minimal area or methods based on the wrapped phase gradient will fail to find the correct branch cut surfaces. We conclude that additional information about the problem under study is therefore needed to produce correct branch cut surfaces that lead to an unwrapped phase distribution with minimum local errors. An example with real data is provided in which downsampled phase contrast magnetic resonance imaging data are successfully unwrapped when the position of the vessel walls and the physical properties of the flowing blood are taken into account.

© 2006 Optical Society of America

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References

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  1. J. M. Huntley and H. Saldner, " Temporal phase-unwrapping algorithm for automated interferogram analysis," Appl. Opt. 32, 3047- 3052 ( 1993).
    [CrossRef] [PubMed]
  2. J. M. Huntley, " Three-dimensional noise-immune phase unwrapping algorithm," Appl. Opt. 40, 3901- 3908 ( 2001).
    [CrossRef]
  3. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).
  4. J. M. Huntley, " Noise-immune phase unwrapping algorithm," Appl. Opt. 28, 3268- 3270 ( 1989).
    [CrossRef] [PubMed]
  5. J. R. Buckland, J. M. Huntley, and S. R. E. Turner, " Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm," Appl. Opt. 34, 5100- 5108 ( 1995).
    [CrossRef] [PubMed]
  6. R. Cusack and N. Papadakis, " New robust 3D phase unwrapping algorithms: application to magnetic field mapping and undistorting echoplanar images," NeuroImage 16, 754- 764 ( 2002).
    [CrossRef] [PubMed]
  7. M. Jenkinson, " Fast, automated, N-dimensional phase-unwrapping algorithm," Magn. Reson. Med. 49, 193- 197 ( 2003).
    [CrossRef] [PubMed]
  8. M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, " Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc. Interface (to be published).
  9. D. J. Bone, " Fourier fringe analysis: the two-dimensional phase unwrapping problem," Appl. Opt. 30, 3627- 3632 ( 1991).
    [CrossRef] [PubMed]
  10. J. Burke, " Application and optimisation of the spatial phase shifting technique in digital speckle interferometry," Ph.D. dissertation (Shaker Verlag, Aachen, Germany, 2001).
  11. S. Chavez, Q. S. Xiang, and L. An, " Understanding phase maps in MRI: a new cutline phase unwrapping method," IEEE Trans. Med. Imag. 21, 966- 977 ( 2002).
    [CrossRef]
  12. O. Marklund, J. M. Huntley, and R. Cusack, " Robust unwrapping algorithm for 3-D phase volumes of arbitrary shape containing knotted phase singularity loops," Research Rep., ISSN 1402-1528 Luleå University of Technology, Sweden, ( 2005).
  13. D. W. McRobbie, E. A. Moore, M. J. Graves, and M. R. Prince, MRI from Picture to Proton (Cambridge U. Press, 2003), pp. 264-269.
  14. K. S. Cunningham, and A. I. Gotlieb, " The role of shear stress in the pathogenesis of atherosclerosis," Lab. Investig. 85, 9- 23 ( 2005).
    [CrossRef]
  15. S. Han, O. Marseille, C. Gehlen, and B. Blümich, " Rheology of blood by NMR," J. Magn. Reson. 152, 87- 94 ( 2001).
    [CrossRef] [PubMed]
  16. Y. -L. Hu, W. J. Rogers, D. A. Coast, C. M. Kramer, and N. Reichek, " Vessel boundary extraction based on a global and local deformable physical model with variable stiffness," Magn. Reson. Imaging 16, 943- 951 ( 1998).
    [CrossRef] [PubMed]
  17. K. C. Wang, R. W. Dutton, and C. A. Taylor, " Geometric image segmentation and image-based model construction for computational hemodynamics," IEEE Eng. Med. Biol. Mag. 18, 33- 39 ( 1999).
    [CrossRef] [PubMed]
  18. Y. Zhang and B. Tabarrok, " Generation of surfaces via equilibrium forces," Comput. Struct. 70, 599- 613 ( 1999).
    [CrossRef]
  19. J. S. Brew and W. J. Lewis, " Computational form-finding of tension membrane structures. Non-finite element approaches. 1-3," Int. J. Numer. Methods Eng. 56, 651- 697 ( 2003).
    [CrossRef]

2005

K. S. Cunningham, and A. I. Gotlieb, " The role of shear stress in the pathogenesis of atherosclerosis," Lab. Investig. 85, 9- 23 ( 2005).
[CrossRef]

2003

J. S. Brew and W. J. Lewis, " Computational form-finding of tension membrane structures. Non-finite element approaches. 1-3," Int. J. Numer. Methods Eng. 56, 651- 697 ( 2003).
[CrossRef]

M. Jenkinson, " Fast, automated, N-dimensional phase-unwrapping algorithm," Magn. Reson. Med. 49, 193- 197 ( 2003).
[CrossRef] [PubMed]

2002

R. Cusack and N. Papadakis, " New robust 3D phase unwrapping algorithms: application to magnetic field mapping and undistorting echoplanar images," NeuroImage 16, 754- 764 ( 2002).
[CrossRef] [PubMed]

S. Chavez, Q. S. Xiang, and L. An, " Understanding phase maps in MRI: a new cutline phase unwrapping method," IEEE Trans. Med. Imag. 21, 966- 977 ( 2002).
[CrossRef]

2001

J. M. Huntley, " Three-dimensional noise-immune phase unwrapping algorithm," Appl. Opt. 40, 3901- 3908 ( 2001).
[CrossRef]

S. Han, O. Marseille, C. Gehlen, and B. Blümich, " Rheology of blood by NMR," J. Magn. Reson. 152, 87- 94 ( 2001).
[CrossRef] [PubMed]

1999

K. C. Wang, R. W. Dutton, and C. A. Taylor, " Geometric image segmentation and image-based model construction for computational hemodynamics," IEEE Eng. Med. Biol. Mag. 18, 33- 39 ( 1999).
[CrossRef] [PubMed]

Y. Zhang and B. Tabarrok, " Generation of surfaces via equilibrium forces," Comput. Struct. 70, 599- 613 ( 1999).
[CrossRef]

1998

Y. -L. Hu, W. J. Rogers, D. A. Coast, C. M. Kramer, and N. Reichek, " Vessel boundary extraction based on a global and local deformable physical model with variable stiffness," Magn. Reson. Imaging 16, 943- 951 ( 1998).
[CrossRef] [PubMed]

1995

1993

1991

1989

An, L.

S. Chavez, Q. S. Xiang, and L. An, " Understanding phase maps in MRI: a new cutline phase unwrapping method," IEEE Trans. Med. Imag. 21, 966- 977 ( 2002).
[CrossRef]

Beauregard, D. A.

M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, " Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc. Interface (to be published).

Blümich, B.

S. Han, O. Marseille, C. Gehlen, and B. Blümich, " Rheology of blood by NMR," J. Magn. Reson. 152, 87- 94 ( 2001).
[CrossRef] [PubMed]

Bone, D. J.

Brew, J. S.

J. S. Brew and W. J. Lewis, " Computational form-finding of tension membrane structures. Non-finite element approaches. 1-3," Int. J. Numer. Methods Eng. 56, 651- 697 ( 2003).
[CrossRef]

Buckland, J. R.

Burke, J.

J. Burke, " Application and optimisation of the spatial phase shifting technique in digital speckle interferometry," Ph.D. dissertation (Shaker Verlag, Aachen, Germany, 2001).

Chavez, S.

S. Chavez, Q. S. Xiang, and L. An, " Understanding phase maps in MRI: a new cutline phase unwrapping method," IEEE Trans. Med. Imag. 21, 966- 977 ( 2002).
[CrossRef]

Coast, D. A.

Y. -L. Hu, W. J. Rogers, D. A. Coast, C. M. Kramer, and N. Reichek, " Vessel boundary extraction based on a global and local deformable physical model with variable stiffness," Magn. Reson. Imaging 16, 943- 951 ( 1998).
[CrossRef] [PubMed]

Cunningham, K. S.

K. S. Cunningham, and A. I. Gotlieb, " The role of shear stress in the pathogenesis of atherosclerosis," Lab. Investig. 85, 9- 23 ( 2005).
[CrossRef]

Cusack, R.

R. Cusack and N. Papadakis, " New robust 3D phase unwrapping algorithms: application to magnetic field mapping and undistorting echoplanar images," NeuroImage 16, 754- 764 ( 2002).
[CrossRef] [PubMed]

O. Marklund, J. M. Huntley, and R. Cusack, " Robust unwrapping algorithm for 3-D phase volumes of arbitrary shape containing knotted phase singularity loops," Research Rep., ISSN 1402-1528 Luleå University of Technology, Sweden, ( 2005).

M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, " Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc. Interface (to be published).

Dutton, R. W.

K. C. Wang, R. W. Dutton, and C. A. Taylor, " Geometric image segmentation and image-based model construction for computational hemodynamics," IEEE Eng. Med. Biol. Mag. 18, 33- 39 ( 1999).
[CrossRef] [PubMed]

Gehlen, C.

S. Han, O. Marseille, C. Gehlen, and B. Blümich, " Rheology of blood by NMR," J. Magn. Reson. 152, 87- 94 ( 2001).
[CrossRef] [PubMed]

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Gotlieb, A. I.

K. S. Cunningham, and A. I. Gotlieb, " The role of shear stress in the pathogenesis of atherosclerosis," Lab. Investig. 85, 9- 23 ( 2005).
[CrossRef]

Graves, M. J.

M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, " Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc. Interface (to be published).

D. W. McRobbie, E. A. Moore, M. J. Graves, and M. R. Prince, MRI from Picture to Proton (Cambridge U. Press, 2003), pp. 264-269.

Han, S.

S. Han, O. Marseille, C. Gehlen, and B. Blümich, " Rheology of blood by NMR," J. Magn. Reson. 152, 87- 94 ( 2001).
[CrossRef] [PubMed]

Hu, Y. -L.

Y. -L. Hu, W. J. Rogers, D. A. Coast, C. M. Kramer, and N. Reichek, " Vessel boundary extraction based on a global and local deformable physical model with variable stiffness," Magn. Reson. Imaging 16, 943- 951 ( 1998).
[CrossRef] [PubMed]

Huntley, J. M.

J. M. Huntley, " Three-dimensional noise-immune phase unwrapping algorithm," Appl. Opt. 40, 3901- 3908 ( 2001).
[CrossRef]

J. R. Buckland, J. M. Huntley, and S. R. E. Turner, " Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm," Appl. Opt. 34, 5100- 5108 ( 1995).
[CrossRef] [PubMed]

J. M. Huntley and H. Saldner, " Temporal phase-unwrapping algorithm for automated interferogram analysis," Appl. Opt. 32, 3047- 3052 ( 1993).
[CrossRef] [PubMed]

J. M. Huntley, " Noise-immune phase unwrapping algorithm," Appl. Opt. 28, 3268- 3270 ( 1989).
[CrossRef] [PubMed]

M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, " Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc. Interface (to be published).

O. Marklund, J. M. Huntley, and R. Cusack, " Robust unwrapping algorithm for 3-D phase volumes of arbitrary shape containing knotted phase singularity loops," Research Rep., ISSN 1402-1528 Luleå University of Technology, Sweden, ( 2005).

Jenkinson, M.

M. Jenkinson, " Fast, automated, N-dimensional phase-unwrapping algorithm," Magn. Reson. Med. 49, 193- 197 ( 2003).
[CrossRef] [PubMed]

Kramer, C. M.

Y. -L. Hu, W. J. Rogers, D. A. Coast, C. M. Kramer, and N. Reichek, " Vessel boundary extraction based on a global and local deformable physical model with variable stiffness," Magn. Reson. Imaging 16, 943- 951 ( 1998).
[CrossRef] [PubMed]

Lewis, W. J.

J. S. Brew and W. J. Lewis, " Computational form-finding of tension membrane structures. Non-finite element approaches. 1-3," Int. J. Numer. Methods Eng. 56, 651- 697 ( 2003).
[CrossRef]

Marklund, O.

O. Marklund, J. M. Huntley, and R. Cusack, " Robust unwrapping algorithm for 3-D phase volumes of arbitrary shape containing knotted phase singularity loops," Research Rep., ISSN 1402-1528 Luleå University of Technology, Sweden, ( 2005).

M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, " Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc. Interface (to be published).

Marseille, O.

S. Han, O. Marseille, C. Gehlen, and B. Blümich, " Rheology of blood by NMR," J. Magn. Reson. 152, 87- 94 ( 2001).
[CrossRef] [PubMed]

McRobbie, D. W.

D. W. McRobbie, E. A. Moore, M. J. Graves, and M. R. Prince, MRI from Picture to Proton (Cambridge U. Press, 2003), pp. 264-269.

Moore, E. A.

D. W. McRobbie, E. A. Moore, M. J. Graves, and M. R. Prince, MRI from Picture to Proton (Cambridge U. Press, 2003), pp. 264-269.

Papadakis, N.

R. Cusack and N. Papadakis, " New robust 3D phase unwrapping algorithms: application to magnetic field mapping and undistorting echoplanar images," NeuroImage 16, 754- 764 ( 2002).
[CrossRef] [PubMed]

Prince, M. R.

D. W. McRobbie, E. A. Moore, M. J. Graves, and M. R. Prince, MRI from Picture to Proton (Cambridge U. Press, 2003), pp. 264-269.

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Reichek, N.

Y. -L. Hu, W. J. Rogers, D. A. Coast, C. M. Kramer, and N. Reichek, " Vessel boundary extraction based on a global and local deformable physical model with variable stiffness," Magn. Reson. Imaging 16, 943- 951 ( 1998).
[CrossRef] [PubMed]

Rogers, W. J.

Y. -L. Hu, W. J. Rogers, D. A. Coast, C. M. Kramer, and N. Reichek, " Vessel boundary extraction based on a global and local deformable physical model with variable stiffness," Magn. Reson. Imaging 16, 943- 951 ( 1998).
[CrossRef] [PubMed]

Saldner, H.

Salfity, M. F.

M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, " Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc. Interface (to be published).

Tabarrok, B.

Y. Zhang and B. Tabarrok, " Generation of surfaces via equilibrium forces," Comput. Struct. 70, 599- 613 ( 1999).
[CrossRef]

Taylor, C. A.

K. C. Wang, R. W. Dutton, and C. A. Taylor, " Geometric image segmentation and image-based model construction for computational hemodynamics," IEEE Eng. Med. Biol. Mag. 18, 33- 39 ( 1999).
[CrossRef] [PubMed]

Turner, S. R. E.

Wang, K. C.

K. C. Wang, R. W. Dutton, and C. A. Taylor, " Geometric image segmentation and image-based model construction for computational hemodynamics," IEEE Eng. Med. Biol. Mag. 18, 33- 39 ( 1999).
[CrossRef] [PubMed]

Xiang, Q. S.

S. Chavez, Q. S. Xiang, and L. An, " Understanding phase maps in MRI: a new cutline phase unwrapping method," IEEE Trans. Med. Imag. 21, 966- 977 ( 2002).
[CrossRef]

Zhang, Y.

Y. Zhang and B. Tabarrok, " Generation of surfaces via equilibrium forces," Comput. Struct. 70, 599- 613 ( 1999).
[CrossRef]

Appl. Opt.

Comput. Struct.

Y. Zhang and B. Tabarrok, " Generation of surfaces via equilibrium forces," Comput. Struct. 70, 599- 613 ( 1999).
[CrossRef]

IEEE Eng. Med. Biol. Mag.

K. C. Wang, R. W. Dutton, and C. A. Taylor, " Geometric image segmentation and image-based model construction for computational hemodynamics," IEEE Eng. Med. Biol. Mag. 18, 33- 39 ( 1999).
[CrossRef] [PubMed]

IEEE Trans. Med. Imag.

S. Chavez, Q. S. Xiang, and L. An, " Understanding phase maps in MRI: a new cutline phase unwrapping method," IEEE Trans. Med. Imag. 21, 966- 977 ( 2002).
[CrossRef]

Int. J. Numer. Methods Eng.

J. S. Brew and W. J. Lewis, " Computational form-finding of tension membrane structures. Non-finite element approaches. 1-3," Int. J. Numer. Methods Eng. 56, 651- 697 ( 2003).
[CrossRef]

J. Magn. Reson.

S. Han, O. Marseille, C. Gehlen, and B. Blümich, " Rheology of blood by NMR," J. Magn. Reson. 152, 87- 94 ( 2001).
[CrossRef] [PubMed]

Lab. Investig.

K. S. Cunningham, and A. I. Gotlieb, " The role of shear stress in the pathogenesis of atherosclerosis," Lab. Investig. 85, 9- 23 ( 2005).
[CrossRef]

Magn. Reson. Imaging

Y. -L. Hu, W. J. Rogers, D. A. Coast, C. M. Kramer, and N. Reichek, " Vessel boundary extraction based on a global and local deformable physical model with variable stiffness," Magn. Reson. Imaging 16, 943- 951 ( 1998).
[CrossRef] [PubMed]

Magn. Reson. Med.

M. Jenkinson, " Fast, automated, N-dimensional phase-unwrapping algorithm," Magn. Reson. Med. 49, 193- 197 ( 2003).
[CrossRef] [PubMed]

NeuroImage

R. Cusack and N. Papadakis, " New robust 3D phase unwrapping algorithms: application to magnetic field mapping and undistorting echoplanar images," NeuroImage 16, 754- 764 ( 2002).
[CrossRef] [PubMed]

Other

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, " Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc. Interface (to be published).

J. Burke, " Application and optimisation of the spatial phase shifting technique in digital speckle interferometry," Ph.D. dissertation (Shaker Verlag, Aachen, Germany, 2001).

O. Marklund, J. M. Huntley, and R. Cusack, " Robust unwrapping algorithm for 3-D phase volumes of arbitrary shape containing knotted phase singularity loops," Research Rep., ISSN 1402-1528 Luleå University of Technology, Sweden, ( 2005).

D. W. McRobbie, E. A. Moore, M. J. Graves, and M. R. Prince, MRI from Picture to Proton (Cambridge U. Press, 2003), pp. 264-269.

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Figures (13)

Fig. 1
Fig. 1

(a) A wrapped phase map (black and white represent, respectively, −π and +π) containing a dipole pair (points 1 and 2) results in a path-dependent unwrapped phase at point Q with respect to point P. (Reprinted, by permission, from Ref. 1). (b) A dipole pair occurs on a 2D surface intersected by a phase singularity loop. (Reprinted, by permission, from Ref. 2).

Fig. 2
Fig. 2

Simple example of a C-shaped 3D ambiguous loop. Two of the possible branch cut surfaces are shown shaded in (a) and (b).

Fig. 3
Fig. 3

Two possible circumstances in two dimensions in which a dipole pair appears owing to undersampling. The continuous phase maps are schematically represented by contour maps, whose isophase lines are spaced by π. The grids represent sampling of the phase maps, with the intersection of vertical and horizontal grid lines being the sampled points. In (a) a straight-line branch cut is appropriate. In (b) a nonstraight branch cut is needed.

Fig. 4
Fig. 4

(a) True phase ϕ u. (b) True wrapped phase ϕ w = U u ]. (c) Downsampled phase ϕ ws with a dipole pair (s 1, s 2) and a straight-line branch cut. (d) Unwrapping ϕ ws by use of the straight-line branch cut produces a localized error that extends along the black region (from the location of the ideal branch cut to the location of the straight-line branch cut). (e) Magnitude of the phase gradient of ϕ w. (e) Magnitude of the phase gradient of ϕ ws.

Fig. 5
Fig. 5

The same PSL can be the result of different undersampling conditions, and therefore the appropriate branch cut surface does not depend on the loop geometry alone. (a) Some of the 2D xy slices of an undersampled wrapped phase volume and the PSL together with its correct branch surface. (b) A different undersampled wrapped phase volume with exactly the same PSL as in (a); however, the branch surface needed to prevent unwrapping errors is entirely different.

Fig. 6
Fig. 6

Example of minimum area surface resulting in local unwrapping errors. (a) Some of the 2D xy slices of an undersampled wrapped phase volume and the PSL present in it, with the minimum area branch cut surface. The resultant unwrapped phase is equal to the wrapped phase. (b) Some xy slices of the true unwrapped phase volume and the PSL with a different surface that follows regions of undersampling. The resultant unwrapped phase volume gives the correct unwrapped phase values.

Fig. 7
Fig. 7

Effect of a finite-sized detector on the PSL of Fig. 5(b): (a) without and (b) with added noise.

Fig. 8
Fig. 8

(a) Phase singularity loops and their branch surfaces that appear in the first three frames of wrapped phase volume ϕ wt 3 in the ascending aorta. (b) Unwrapped phase corresponding to the frames shown in (a).

Fig. 9
Fig. 9

Velocity values for the pixel that has the maximum velocity inside the ascending aorta, shown for the first 35 frames of full-resolution wrapped phase volume ϕ w. The frames taken for the temporally downsampled volumes are also shown.

Fig. 10
Fig. 10

(a) Phase singularity loops and their branch surfaces that appear in the first three frames of temporally downsampled wrapped phase volume ϕ wt 5 in the ascending aorta. Note that the larger loop has both horizontal and vertical components. (b) Same phase subvolume as in (a) with the larger loop surface modified to bring all vertical portions of the loop toward the vessel wall. (c) Unwrapped phase corresponding to (a); a small unwrapping error appears on frame 2. (d) Unwrapped phase corresponding to (b). (e) ∇2 v corresponding to frame 2 of (c) (σ L = 9.6 s-1 cm-1, inside the vessel). (f) ∇2 v corresponding to frame 2 of (d) (σ L = 7.4 s−1 cm−1 inside the vessel).

Fig. 11
Fig. 11

(a) Phase singularity loops and their branch surfaces that appear in the first three frames of temporally downsampled wrapped phase volume ϕwt6 in the ascending aorta. (b) Same phase subvolume as in (a) with the larger loop surface modified to have horizontal patches in horizontal portions of the loop and to bring all vertical portions of the loop toward the vessel wall. (c) Unwrapped phase corresponding to (a); there is an unwrapping error on frame 2. (d) Unwrapped phase corresponding to (b). (e) ∇2 v corresponding to frame 2 of (c) (σ L = 12.6 s-1 cm-1, inside the vessel). (f) ∇2 v corresponding to frame 2 of (d) (σ L = 7.4 s−1 cm−1, inside the vessel).

Fig. 12
Fig. 12

(a) One frame of full resolution volume ϕ w. (b) Same frame of spatially downsampled phase volume ϕws2. White pixels indicate where the velocity difference between neighboring pixels is below V enc.

Fig. 13
Fig. 13

(a) Phase singularity loops and their branch surfaces that appear around frame 9 of spatially undersampled wrapped phase volume ϕws2 in the ascending aorta. (b) Same phase subvolume as in (a) with the larger loop surface modified to bring vertical portions of the loop toward the vessel wall. (c) Unwrapped phase corresponding to (a). (d) Unwrapped phase corresponding to (b). (e) ∇2 v corresponding to (c) (σ L = 15.8 s-1 cm-1, inside the vessel). (f) ∇2 v corresponding to frame 2 of (d) (σ L = 13.3 s−1 cm−1 inside the vessel).

Equations (140)

Equations on this page are rendered with MathJax. Learn more.

( π , π ]
L p
V enc
ϕ w
( x , y ) ,
( x + 1 , y ) ,
( x + 1 , y + 1 ) ,
( x , y + 1 )
s = NINT { [ ϕ ( x + 1 , y ) ϕ ( x , y ) ] / 2 π } + NINT { [ ϕ ( x + 1 , y + 1 ) ϕ ( x + 1 , y ) ] / 2 π } + NINT { [ ϕ ( x , y + 1 ) ϕ ( x + 1 , y + 1 ) ] / 2 π } + NINT { [ ϕ ( x , y ) ϕ ( x , y + 1 ) ] / 2 π } ,
1
2 × 2 × 2
ϕ u
ϕ w
U [ ϕ u ]
U [ ϕ ]
ϕ 2 π NINT
[ ϕ / 2 π ]
ϕ u
ϕ w
ϕ u s
ϕ w s
ϕ u s
| Δ ϕ u s | < π
| Δ ϕ u s | > π
ϕ w
ϕ w s
ϕ w s
s 1
s 2
| Δ ϕ u s | > π
s 1
s 2
256 × 256
ϕ u
1.5 π
63 × 63
21 × 21
1 / e 2
1 / e 2
ϕ w
ϕ w s
( s 1 , s 2 )
ϕ w s
ϕ w
| Δ ϕ u s | > π
ϕ w s
128 × 128 × 128
ϕ u
31 × 31 × 31
1.5 π
21 × 21 × 21
1 / e 2
1 / e 2
| Δ ϕ u s | > π
| Δ ϕ u s | < π
ϕ w
ϕ w s
ϕ w s
| Δ ϕ u s | > π
ϕ w s
1 / e 2
| Δ ϕ u s | < π
| Δ ϕ u s | > π
( < 10 )
ϕ u
0.1   rad
ϕ w s
ϕ u
v ( ϕ ) = ϕ V enc π .
V enc
V enc
V enc
ρ v t = P + μ ( 2 v x 2 + 2 v y 2 ) ,
P
v ( x , y ) = Δ P R 2 4 μ [ 1 ( x 2 + y 2 ) R 2 ] .
v ( x , y )
2 v = 2 v / x 2 + 2 v / y 2
2 v
σ L
σ L
δ t
a p
| a p | δ t V enc .
| v x | p δ x V enc ,
| v y | p δ y V enc ,
δ x   and   δ y
ϕ w
1.5   T
V enc = 50   cm   s 1
280 × 280   mm
5.0   mm
256 × 128
31.3   kHz
30 °
4   ms
TR,   9   ms
18   ms
ϕ w
ϕ wt 3 ,
ϕ wt 5 ,
ϕ wt 6 .
ϕ w
ϕ ws 2
ϕ w
ϕ u
ϕ u
ϕ ut 3 ,
ϕ ut 5 ,
ϕ ut 6 ,
ϕ us 2
ϕ wt 3
ϕ u
51   cm   s 1
V enc
ϕ wt 3
ϕ ut 3
0   cm   s 1
ϕ wt 5
75   cm   s 1
ϕ ut 5
0.28   cm   s 1
2 v
2 v
3 × 3
[ 1 2 1 1 2 1 1 2 1 ] ,        [ 1 1 1 2 2 2 1 1 1 ] ,
ϕ ut 5
σ L
ϕ wt 6
ϕ wt 6
ϕ ut 6
0.81   cm   s 1
2 v
σ L
ϕ w
V enc
ϕ ws 2
ϕ ws 2
ϕ us 2
σ L
30

Metrics